The "natural" movement (or movement learned and embodied so well it becomes natural) of sword and spear/staff seem very different. Does emphasizing sword over jo suburi, for example, make a difference in empty-hand technique? Does a particular emphasis affect how internal body connection is cultivated and trained?

Ellis Amdur's historical inquiry into the course of Ueshiba Morihei's personal development in weapons work is interesting. I wonder how people training today with both sword and spear/staff find the weapons training with respect to its usefulness in building internal body skill.

Sword and Spear/Staff work are inter-related, or at least I've found so in my own training.
The internal "saggital" plane work done with spear/staff is directly applicable to sword work, and its the stepping stone to examining other components of internal work.

Ever see some Japanese weapons practicioners hold their naginata above their heads (raised to the "heaven" position)? That's not a coincidence. Nor is the direct corallary in the sword, where all you're doing is moving (inside yourself) from "heaven" to "earth". IE dropping the dantien or whatever you want to call it. There's all sorts of other winding, coiling, storing etc that you can do, but with a weapon in your hands, it makes it that much more demanding to do internal practice properly.

Both Sagawa and Tohei mention that " You see plenty of people out there swinging a bokken like idiots a thousand times a day. All they're doing is building up a thousand differenent bad habits. "
(The only part where they differ is where Sagawa goes on to make the crack that people like are a) too stupid to practice MAs, and b) they should be cracked on the head to be given a reality check )

I use weapons work to train my body all the time, and it definitely has a direct carry over to my empty hands work, when I spar with mma kids.

Said properties/harmonies are put into weapons work, such as spear or cutting. Test and train these realizations with the weapons, which again help the body to realize other factors

Said factors are then put into Solo Work(sans weapons) etc etc.
Its a never ending feedback loop.

Partner work and sparring fit into this feedback loop of course, but they're more of a barometer than a developer if you ask me.

My two cents anyways.

PS
Practicing with any kind of weapon should reveal any "imbalances" you have in your body movement/alignments. Which is why long weapons that exert leverage on the body are used since you can't use "muscle" to wield them. Practicing on a 6 foot bokken quickly whips the bodies alignments into shape